Algebraic-group factorisation algorithm

Algebraic-group factorisation algorithms are algorithms for factoring an integer N by working in an algebraic group defined modulo N whose group structure is the direct sum of the 'reduced groups' obtained by performing the equations defining the group arithmetic modulo the unknown prime factors p₁, p₂, ... By the Chinese remainder theorem, arithmetic modulo N corresponds to arithmetic in all the reduced groups simultaneously.

The aim is to find an element which is not the identity of the group modulo N, but is the identity modulo one of the factors, so a method for recognising such one-sided identities is required. In general, one finds them by performing operations that move elements around and leave the identities in the reduced groups unchanged. Once the algorithm finds a one-sided identity all future terms will also be one-sided identities, so checking periodically suffices.

Computation proceeds by picking an arbitrary element x of the group modulo N and computing a large and smooth multiple Ax of it; if the order of at least one but not all of the reduced groups is a divisor of A, this yields a factorisation. It need not be a prime factorisation, as the element might be an identity in more than one of the reduced groups.

Generally, A is taken as a product of the primes below some limit K, and Ax is computed by successive multiplication of x by these primes; after each multiplication, or every few multiplications, the check is made for a one-sided identity.